Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{7r^2 + 7r - 84}{-r^3 + 11r^2 - 24r}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {7(r^2 + r - 12)} {-r(r^2 - 11r + 24)} $ $ n = -\dfrac{7}{r} \cdot \dfrac{r^2 + r - 12}{r^2 - 11r + 24} $ Next factor the numerator and denominator. $ n = - \dfrac{7}{r} \cdot \dfrac{(r - 3)(r + 4)}{(r - 3)(r - 8)}$ Assuming $r \neq 3$ , we can cancel the $r - 3$ $ n = - \dfrac{7}{r} \cdot \dfrac{r + 4}{r - 8}$ Therefore: $ n = \dfrac{ -7(r + 4)}{ r(r - 8)}$, $r \neq 3$